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(1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a topology on A. (b) Show that if 7 is Hausdorff, then T|A is Hausdorff.

(1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a topology on A. (b) Show that if 7 is Hausdorff, then T|A is Hausdorff.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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(1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a topology on A. (b) Show that if 7 is Hausdorff, then T|A is Hausdorff.
Transcribed Image Text:(1) Let (X, 7) be a topological space and let A CX. Let TA= {UNA: U ET}. (a) Show that TA is a topology on A. (b) Show that if 7 is Hausdorff, then T|A is Hausdorff.
Expert Solution
Step 1: Definition

Topology: Let τ be a collection of subsets of a nonempty set X. Then τ is called a topology if this collection has the properties

1) Empty set ϕ and X are in τ.

2) A,B∈τ then A∩B∈τ.

3) For any arbitrary collection Aα∈τ, α is an arbitrary index set then ∪αAα∈τ.

Elements of τ are called open set with respect to the topology τ.


Hausdorff topology: Let τ be a topology on a set X. If every two distinct points p,q∈X there exists two disjoint open sets Up,Uq containing p,q respectively.

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